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1: 1.6 Vectors and Vector-Valued Functions

§1.6 Vectors and Vector-Valued Functions

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§1.6(i) Vectors

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Unit Vectors

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Cross Product (or Vector Product)

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§1.6(ii) Vectors: Alternative Notations

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2: 37.17 Hermite Polynomials on ${ℝ}^d$

… ►On ${ℝ}^d$ consider the weight function $\exp \left(-{\Vert 𝐱\Vert }^2\right)$ and the corresponding inner product … ►

37.17.5 $$\sum _{|𝝂|=n}\frac{H_{𝝂}(𝐱)}{𝝂!}{𝐲}^{𝝂}=\frac{1}{n!}{H}_n\left(⟨𝐱,𝐲⟩\right),$$ $\Vert 𝐲\Vert =1$,

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and $𝐲$: vector in $d$-dimensional Euclidean space

Proof sketch:

Take $\Vert 𝐲\Vert =1$ in (37.17.4).

Permalink:

http://dlmf.nist.gov/37.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(i), §37.17(i), §37.17, Part 37.III and Ch.37

… ►Specialization in §37.13(i) of the rotation invariant weight function to $W(𝐱)=\exp \left(-{\Vert 𝐱\Vert }^2\right)$ gives for the corresponding OPs that … ►

37.17.11 $${𝐏}_z(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n(𝐱,𝐲)z^n=\frac{1}{{\left(1-z^2\right)}^{\frac{d}{2}}}\exp \left(\frac{2z⟨𝐱,𝐲⟩-z^2({\Vert 𝐱\Vert }^2+{\Vert 𝐲\Vert }^2)}{1-z^2}\right),$$ .

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Symbols:

$\exp z$: exponential function, $⟨𝐮,𝐯⟩$: inner product over vectors, $(a,b)$: open interval, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $z$: complex number and ${𝐑}_n$: reproducing kernel

Proof sketch:

Use (18.18.28).

Permalink:

http://dlmf.nist.gov/37.17.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(iv), §37.17, Part 37.III and Ch.37

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§37.17(vi) Hermite Polynomials for Weight Function ${\mathrm{e}}^{-⟨𝐀𝐱,𝐱⟩}$

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3: 1.2 Elementary Algebra

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§1.2(v) Matrices, Vectors, Scalar Products, and Norms

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Row and Column Vectors

… ►and the corresponding transposed row vector of length $n$ is … ►

Vector Norms

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1.2.52 $$\Vert 𝐮+𝐯\Vert \le \Vert 𝐮\Vert +\Vert 𝐯\Vert .$$

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Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E52

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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4: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►These are OPs on the bounded cone ${𝕍}^{d+1}=\{(𝐱,t)\mid \Vert 𝐱\Vert \le t,t\in [0,1],𝐱\in {ℝ}^d\}$ associated to the Jacobi weight function … ►where ${\mathrm{\Delta }}_{𝐱}$ and ${\nabla }_{𝐱}$ are the Laplace operator and the gradient vector in the variable $𝐱$. … ►and $x_{d+1}=\sqrt{t^2-{\Vert 𝐱\Vert }^2}$ and $y_{d+1}=\sqrt{s^2-{\Vert 𝐲\Vert }^2}$; moreover, if either $\mu =-1$, $\gamma =-\frac{1}{2}$, and/or $d=2$, the identity (37.18.9) holds under the limit relation (37.14.14). … ►These are OPs on the unbounded cone ${𝕍}^{d+1}=\{(𝐱,t)\mid \Vert 𝐱\Vert \le t,t\in {ℝ}_{+},𝐱\in {ℝ}^d\}$ associated to the Laguerre weight function … ►where ${\mathrm{\Delta }}_{𝐱}$ and ${\nabla }_{𝐱}$ are the Laplace operator and the gradient vector in the variable $𝐱$. …

5: 37.15 Orthogonal Polynomials on the Ball

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37.15.1

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Defines:

${𝔹}^d$: unit ball (locally)

Symbols:

$\in$: element of, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), ${ℝ}^d$: $d$-dimensional Euclidean space and $𝐱$: vector in $d$-dimensional Euclidean space

Permalink:

http://dlmf.nist.gov/37.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(i), §37.15, Part 37.III and Ch.37

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37.15.2 $$W_{\alpha }(𝐱)={\left(1-{\Vert 𝐱\Vert }^2\right)}^{\alpha },$$ $\alpha >-1$,

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Defines:

$W$: weight function in $d$-variables (locally)

Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Referenced by:

§37.15(iii), §37.18(ii), §37.18(iii)

Permalink:

http://dlmf.nist.gov/37.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(i), §37.15, Part 37.III and Ch.37

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37.15.13 $${\left(1-2⟨𝐱,𝐲⟩+{\Vert 𝐲\Vert }^2\right)}^{-\alpha -\frac{1}{2}d}=\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{2^{|𝝂|}{\left(\alpha +\frac{1}{2}d\right)}_{|𝝂|}}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱),$$ $𝐲\in {ℝ}^d$, .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Proved:

Dunkl and Xu (2014, Proposition 5.2.3); There the polynomial $V_{𝝂}^{(\alpha +\frac{1}{2})}$ is denoted by $V_{\alpha }$.(proved)

Referenced by:

(37.15.14)

Permalink:

http://dlmf.nist.gov/37.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

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37.15.15 $${\left({(1-⟨𝐱,𝐲⟩)}^2+{\Vert 𝐲\Vert }^2\left(1-{\Vert 𝐱\Vert }^2\right)\right)}^{-\alpha -\frac{1}{2}}=\sum _{𝝂\in {\mathbb{N}}_0^d}\frac{{(-1)}^{|𝝂|}{\left(2\alpha +1\right)}_{|𝝂|}}{2^{|𝝂|}{\left(\alpha +1\right)}_{|𝝂|}𝝂!}{U}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱){𝐲}^{𝝂},$$ $𝐲\in {ℝ}^d$, .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $\mathbb{N}$: set of all positive integers, $ℝ$: real line, $\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Proved:

Dunkl and Xu (2014, Proposition 5.2.3)(proved)

Permalink:

http://dlmf.nist.gov/37.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

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37.15.18 $${𝐑}_n^{\alpha }(𝐱,𝐲)={\int }_{-1}^1{Z}_n^{\alpha +\frac{1}{2}d}\left(⟨𝐱,𝐲⟩+t\sqrt{1-{\Vert 𝐱\Vert }^2}\sqrt{1-{\Vert 𝐲\Vert }^2}\right)\frac{{\left(1-t^2\right)}^{\alpha -\frac{1}{2}}}{{\int }_{-1}^1{\left(1-t^2\right)}^{\alpha -\frac{1}{2}}dt}dt,$$ $𝐱,𝐲\in {𝔹}^d$, $\alpha >-\frac{1}{2}$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝔹}^d$: unit ball and ${𝐑}_n$: reproducing kernel

Proved:

Dunkl and Xu (2014, Theorem 5.2.8)(proved)

Referenced by:

§37.15(vi), §37.4(i)

Permalink:

http://dlmf.nist.gov/37.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vi), §37.15, Part 37.III and Ch.37

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6: 37.1 Notation

… ► ►►►►

$n$	nonnegative integer.
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$\mathrm{\oplus }$	orthogonal (direct) sum of vector spaces.
$\mathrm{\otimes }$	tensor product of vector spaces.
…

… ► ►►►►

$d$	positive integer, usually $\ge 2$.
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$𝐱,𝐲$	$(x_1,\mathrm{\dots },x_d),(y_1,\mathrm{\dots },y_d)\in {ℝ}^d$.
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$\Vert 𝐱\Vert$	$\sqrt{x_1^2+\mathrm{\cdots }+x_d^2}$ ($𝐱\in {ℝ}^d$).
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7: 37.19 Other Orthogonal Polynomials of $d$ Variables

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37.19.1 $$T_{j}f(𝐱)=\frac{\partial f}{\partial x_j}+\sum _{𝐯\in R_{+}}{\kappa }_{𝐯}\frac{f(𝐱)-f(𝐱{\sigma }_{𝐯})}{⟨𝐱,𝐯⟩}{v}_j,$$

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Symbols:

$\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\partial x$: partial differential of $x$ , $j$: nonnegative integer, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Sources:

Dunkl (1989, Definition 1.3); Dunkl and Xu (2014, Definition 6.4.2)

Permalink:

http://dlmf.nist.gov/37.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

►where $v_{\mathrm{\ell }}$ is the $\mathrm{\ell }$th component of $𝐯$ and $𝐱{\sigma }_{𝐯}$ denotes the reflection $𝐱{\sigma }_{𝐯}=𝐱-2\frac{⟨𝐱,𝐯⟩}{⟨𝐯,𝐯⟩}𝐯.$ These operators commute; that is, $T_{\mathrm{\ell }}{T}_j=T_j{T}_{\mathrm{\ell }}$ for . … ►

37.19.6 $$W_{\kappa ,\mu }(𝐱)=w_{\kappa }(𝐱){\left(1-{\Vert 𝐱\Vert }^2\right)}^{\mu -\frac{1}{2}}$$

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Defines:

$W$: weight function in $d$-variables (locally)

Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²), $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Source:

Dunkl and Xu (2014, Definition 8.1.1)

Permalink:

http://dlmf.nist.gov/37.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(ii), §37.19, Part 37.III and Ch.37

… ►For the radial weight function ${\Vert 𝐱\Vert }^{\alpha }{(1-{\Vert 𝐱\Vert }^2)}^{\mu -\frac{1}{2}}$ ($\mu >-\frac{1}{2}$) on the unit ball, orthogonal polynomials are studied in Xu (2015) and a closed-form formula of the reproducing kernels is established. … ►Orthogonal polynomials for the weight function $w_{\kappa }(𝐱){\mathrm{e}}^{-{\Vert 𝐱\Vert }^2}$ on ${ℝ}^d$ can be defined explicitly and most of §37.17 can be extended to this more general setting. …

8: 1.1 Special Notation

… ► ►►►►►

$x,y$	real variables.
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$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
…
$𝐮$, $𝐯$	column vectors.
${𝐄}_n$	the space of all $n$-dimensional vectors.
…

…

9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space $V$ is called an inner product space if an inner product $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. … ►

1.18.1 $$\Vert v\Vert =\sqrt{⟨v,v⟩},$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $\Vert 𝐯\Vert$: vector norm ($l$²)

Referenced by:

Erratum (V1.2.0) §1.18

Permalink:

http://dlmf.nist.gov/1.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(i), §1.18 and Ch.1

► $V$ becomes a normed linear vector space. If $\Vert v\Vert =1$ then $v$ is normalized. Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. …

10: 21.1 Special Notation

… ► ►►►►►►

$g,h$	positive integers.
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$𝜶,𝜷$	$g$-dimensional vectors, with all elements in $[0,1)$, unless stated otherwise.
$a_j$	$j$th element of vector $𝐚$.
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$𝐚\cdot 𝐛$	scalar product of the vectors $𝐚$ and $𝐛$.
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$S^g$	set of $g$-dimensional vectors with elements in $S$.
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►Lowercase boldface letters or numbers are $g$-dimensional real or complex vectors, either row or column depending on the context. …